I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will know and understand. I do not like to discuss what beauty means...I have now written down a script with 107 such theorems (in my sense) and their proof and I have ideas for another 20. One the one hand I would be very thankful for theorems I have yet not considered. On the other hand I like to see if there is some consensus which theorems with a short proof are beautiful.

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3

This should be big list and community wiki...
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Igor RivinSep 10 '12 at 23:56

It is indeed a very slick proof, but the original question says: "A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will know and understand." How much complex analysis should these students know?
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Yemon ChoiSep 11 '12 at 6:17

Euler's formula $$\mathrm{e}^{i \theta} = \cos \left( \theta \right) + i \sin \left( \theta
\right)$$ when considered as a theorem. From whatever angle you look at it, almost all the proofs are short and extremely beautiful.